The Onset of Cold Advection During Upper-Level Frontogenesis

Chuck Doswell and I have submitted the following manuscript to Quarterly Journal of the Royal Meteorological Society: "Conceptual Models of Upper-Level Frontogenesis in Southwesterly and Northwesterly Flow." You can view a copy of that manuscript here. If you have problems viewing the equations, try this version.

Rotunno et al. (1994) state:

"At day 7 there is a strong departure from this pattern [theta and streamfunction are substantially in phase near the jet center] as the isotherms begin to cross the lines of constant streamfunction just upstream of the trough. . . . it is clear that this development is associated with the descent of high-PV air through the z=6 km level." (p. 3387)

"At day 7 there is dramatic change in the pattern of theta and streamfunction as mentioned above. Previous to this development, the large-scale wave grows with theta and streamfunction mostly in phase. After this development, there is a local change in the pattern of theta and streamfunction such that there is strong cold advection just SW of the trough. This development implies a significant change in the Q-vector distribution: The pattern of theta and streamfunction implies a frontolytical geostrophic horizontal shear that makes Q point toward cold air on the cyclonic shear side of the jet; with Q continuing to point toward warm air on the anticyclonic shear side of the jet. . . ." (pp. 3389-3390).

"In the pre-wave breaking stage, which is the stage considered by almost all past investigators, the chain of cause and effect runs like this: Consider in the first step the flow in Fig. 13a---the isolines of theta and streamfunction are roughly parallel but there is a small amount of geostrophic confluence south, and diffluence north, of a NW-SE line running from ridge to trough. The ageostrophic response indicated by Q is horizontal motion toward that line and sinking motion along it. This implies frontogenetical tilting to the north and frontolytical tilting to the south of that line. As the sinking motion grows in intensity with distance from the ridge to along the line, and in time, higher-theta air from aloft descends past the z= 6 km level; this implies that the isotherms at this level will no longer be roughly parallel to the lines of constant streamfunction but rather will turn sharply near the trough as shown in Fig. 13b." (p. 3390)

Lagrangian formulation of Fs

Calculations of the horizontal and vertical terms in the rotational frontogenesis expression Fs (Keyser et al. 1988; Schultz and Doswell 1999) using Rotunno et al.'s simulation show that the horizontal vorticity term is responsible for the cyclonic rotation of the isentropes relative to the flow both at small amplitude (day 5) and at large amplitude (day 7). The vertical tilting term (Rotunno et al.'s proposed mechanism) actually acts to rotate the isentropes anticyclonically, opposite the observed direction.

Fs FIGURE: diagnostics at day 5

Fs FIGURE: diagnostics at day 7

Figure Caption:

5625-m relative vorticity of total horizontal wind (10-5 s-1, shaded according to scale at bottom of figure), and potential temperature (thin solid lines every 2 K) at day 5 (day 7): (a) Fs vorticity term (every 3 ×10-11 (-10) K m s-1, from -18 to 18 ×10-11 (-10) K m s-1); (b) Fs deformation term (every 3×10-11 (-10) K m s-1, from -18 to 18 ×10-11 (-10) K m s-1) and axes of dilatation of total horizontal wind (10-5 s, scaled according to legend; every other grid point)); (c) Fs vertical tilting term (every 3 ×10-11 (-10) K m s-1, from -18 to 18 ×10-11 (-10) K m s-1); (d) total Fs (every 3 ×10-11 (-10) K m s-1, from -18 to 18×10-11 (-10) K m s-1).
Eulerian derivation of Fs and a

Keyser et al. (1988, p. 764) present the following expression:

Fs = |Ñq| d a
dt
 ,
(1)
where a is the angle between the positive-x axis and the isentrope orientation measured in a counterclockwise direction. Expanding da/dt,
d a
dt
 = a
t
 + u a
x
 + v a
y
 + w a
z
 .
(2)
Solving for a/ t,
a
t
 = |Ñq|-1 Fs- u a
x
 - v a
y
 - w a
z
 .
(3)
Therefore, we can use (3) to determine the factors which cause the rotation of the isentropes at a point (Eulerian formulation). The Eulerian (at a point) rate of change of a is equal to the Lagrangian rate of change of a (atmospheric processes detailed above) plus horizontal advection plus vertical advection.

At both day 5 and day 7, the vertical advection term (panel a) is swamped by the horizontal advection term (panel b), which in turn is swamped by the Lagrangian rotational frontogenesis (panel d) to give a net cyclonic turning (positive a/ t in panel c) in the base of the thermal trough. Thus, the primary process rotating the isotherms is Fs, of which we know that the vorticity is the main contribution.

Fs FIGURE: alpha diagnostics at day 5 (scale factor=7)

Fs FIGURE: alpha diagnostics at day 5 (scale factor=8)

Fs FIGURE: alpha diagnostics at day 7

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If you have any further questions about the research discussed here, or desire a manuscript, please feel free to write to me: david.schultz@noaa.gov.

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